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The Pythagorean Identity
Pythagorean Theorem
In a right triangle, the sum of the squares of the lengths
of the legs is equal to the square of the length of the
hypotenuse.
c
a
b
a 2 + b 2 = c2
Remember from Section 9-2
B
Hypotenuse
Leg
Opposite
A
C
Leg Adjacent
Length of Leg Opposite A
Sine of A =
Length of Hypotenuse
Length of Leg Adjacent A
Cosine of A =
Length of Hypotenuse
The Pythagorean Identity
Identity
An equation that is true for all allowed values of the
variable.
In trigonometry, the Pythagorean Identity states that in
a right triangle:
(sin x)2 + (cos x)2 = 1
(where x is an angle)
Note: (sin x)2 and (cos x)2 are commonly written as
sin2x and cos2x, respectively.
Pythagorean Identity and the 45 Angle
2
sin45 = ---2
B
2
cos45  = ---2
2
1
45
A
1
C
2 2
2 2 2
2
(sin 45 ) + (cos 45 ) = ( )  ( ) =  = 1
2
2
4
4
o 2
o 2
Pythagorean Identity and the 60 Angle
B
3
sin60 = ---2
1
cos60  = ---2
2
60
1
A
3
C
3 2
1 2 3
1
(sin 60 ) + (cos 60 ) = ( )  ( ) =  = 1
2
2
4
4
o 2
o 2
Pythagorean Identity and the 30 Angle
B
1
sin30 = ---2
3
cos30 = ---2
30
2
A
1
3
C
1 2
3 2 1
3
(sin 30 ) + (cos 30 ) = ( )  ( ) =  = 1
2
2
4
4
o 2
o 2
Example 1
Show that (sin B)2 + (cos B)2 = 1
B
13
5
A
12
C
Example 2
Show that (sin C)2 + (cos C)2 = 1
B
20
5
A
C
Example 3
Show that (sin C)2 + (cos C)2 = 1
B
72o
A
C
Homework:
Practice Worksheet:
(Pythagorean Identity)